The Bayesian framework allows one to integrate produc-tion data and static data into a posteriori probability den-sity function for reservoir variables (model parameters).The problem of generating realizations of the reservoirvariables for the assessment of uncertainty in reservoir de-scription or predicted reservoir performance then becomesa problem of sampling this posteriori pdf to obtain a suiteof realizations. Generation of a realization by the ran-domized maximum likelihood method requires the mini-mization of an objective function that includes productiondata misˉt terms and a model misˉt term that arises fromstatic data. Minimization of this objective function withan optimization algorithm is equivalent to the automatichistory matching of production data with a prior modelconstructed from static data providing regularization. Be-cause of the computational cost of computing sensitivitycoe±cients and the need to solve matrix problems involv-ing the covariance matrix for the prior model, this ap-proach has not been applied to problems where the numberof data and number of reservoir model parameters are bothlarge and the forward problem is solved by a conventionalˉnite di?erence simulator.In this work, we illustrate that computational e±-ciency problems can be overcome by using a scaled limitedmemory Broyden-Fletcher-Goldfarb-Shanno (LBFGS) al-gorithm to minimize the objective function, and using ap-proximate computational stencils to approximate the mul-tiplication of a vector by the prior covariance matrix orits inverse. Implementation of the limited memory BFGSmethod requires only the gradient of the objective func-tion which can be obtained from a single solution of theadjoint problem; individual sensitivity coe±cients are notneeded. We apply the overall process to two examples.The ˉrst is a true ˉeld example in which a realization oflog-permeabilities at 26,019 grid blocks is generated by theautomatic history matching of pressure data and the sec-ond is a pseudo-ˉeld example that provides a very roughapproximation to a North Sea reservoir in which a realiza-tion of log-permeabilities at 9750 gridblocks is computedby the automatic history matching of GOR and pressuredata.
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